I think in everyday language they are pretty close. "Dispose of" often has connotations of getting rid of something properly, while "discard" can have connotations of careless disposal, as in just dropping something in the street.
Of course you as a lawyer are aware of the much broader meaning of "dispose of". But I think the average person is unaware of those meanings and therefore doesn't take them into account in choosing between "dispose of" and "discard".
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